Bunyakovsky conjecture: Difference between revisions
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==Statement== | ==Statement== | ||
The conjecture (not yet proved or disproved) states the following: suppose <math>p(x) \in \mathbb{Z}[x]</math>, i.e., <math>p</math> is | The conjecture (not yet proved or disproved) states the following: suppose <math>p(x) \in \mathbb{Z}[x]</math> is irreducible, i.e., <math>p</math> is an irreducible polynomial of degree two or higher with integer coefficients. Consider the set: | ||
<math>S = \{ p(n) \mid n \in \mathbb{N} \}</math> | <math>S = \{ p(n) \mid n \in \mathbb{N} \}</math> | ||
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* [[Dirichlet's theorem on primes in arithmetic progressions]] is the analogous statement for polynomials of degree one. | * [[Dirichlet's theorem on primes in arithmetic progressions]] is the analogous statement for polynomials of degree one. | ||
* [[Cohn's irreducibility criterion]] is a converse of sorts, which says that a polynomial with nonnegative coefficients takes a prime value at any natural number greater than all coefficients, then the polynomial is irreducible. | |||
===Related conjectures=== | ===Related conjectures=== | ||
* [[Bateman-Horn conjecture]] is a stronger conjecture that also makes assertions about the ''frequency'' of primes in the set of values taken by a polynomial. | * [[Bateman-Horn conjecture]] is a stronger conjecture that also makes assertions about the ''frequency'' of primes in the set of values taken by a polynomial. | ||
Latest revision as of 00:48, 3 July 2012
Statement
The conjecture (not yet proved or disproved) states the following: suppose is irreducible, i.e., is an irreducible polynomial of degree two or higher with integer coefficients. Consider the set:
![{\displaystyle p(x)\in \mathbb {Z} [x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb44fd7ae38db591820a0c9c939c1a7f3d0038a)
Then, one of these two cases must hold:
- The greatest common divisor of all the elements of is greater than 1, i.e., all elements of have a nontrivial common factor.
- contains infinitely many prime numbers.
Note that the first case occurs if and only if the polynomial can be written as (a positive integer greater than 1) times (an integer-valued polynomial).
Related facts and conjectures
Related known facts
- Dirichlet's theorem on primes in arithmetic progressions is the analogous statement for polynomials of degree one.
- Cohn's irreducibility criterion is a converse of sorts, which says that a polynomial with nonnegative coefficients takes a prime value at any natural number greater than all coefficients, then the polynomial is irreducible.
Related conjectures
- Bateman-Horn conjecture is a stronger conjecture that also makes assertions about the frequency of primes in the set of values taken by a polynomial.